x^0 and x^1 Power

Fresco,

I did some thinking.... then I did some digging, about the term "there are no straight lines in nature". It is simply wrong and any smart kid will realize this from her own experience.

There are plenty of examples of straight lines in classical science (the science that anyone who hasn't been to graduate school will be exposed to).

1) Cleavage planes in crystals (the thing that makes gemstones have flat sides).
2) How things fall.
3) Newtons 1st law.
4) There are structures in living cells that are so straight, biologists use the term tensegrity to describe them.

(and there are many more)

You don't get much help in Relativity (or QM). Don't let the term "curved space" fool you... there are plenty of straight lines in either (you don't study Relativity without spending a lot of time with Vector fields.

I looked into the history of the phrase-- it doesn't even come from a scientist. It was a marketing gimmick coined by Antoni Gaudi; a Spanish Architect who used it to promote his style of designing buildings (which apparently didn't have many straight lines).

Somehow it entered popular culture. It still doesn't teach anything about science.

It is a bad thing to pass off unscientific ideas as science - particularly in a science classroom.

American society has big problem because many Americans can't distinguish between real science (measurable facts) from popular myths or vague concepts that aren't well understood. Propagating these myths in a science classroom is indefensible

I am bemused.

Firstly, I had never heard of "the no straight lines concept" except as an apocryphal sign of "extraterrestrial intelligence". (e.g finding geometric lines on the Moon).

Secondly, you don't seem to have considered microscopic viewpoints for crystals or third party views from different reference frames of apparantly "straight trajectories". In essence you appear to unaware of the problems of definition of "straight" which have led to post-relativistic modification of rules such as "light travels in a direction which minimises its transit time". Indeed the phrase "naive realism" springs to mind with respect to your conception of "physical facts".

I have no problems at all with my former activities as a relatively (ho ho) popular and successful teacher (so my former students tell me) . One might suggest that those who rationalise their own leaving of teaching are no position to criticise those who remain.

BTW,

This is not a philosophy thread, but if it were, I would suggest you look at epistemological problems involving:
active versus passive perception (KANT),
"explanation" as prediction and control (CAPRA),
the status of axioms (GODEL's incompleteness theorem)

A simpler (ergo, more teachable) way to think of these two powers is something called the "empty product" and the concept of the multiplicative identity, respectively:

x^n = 1 * x(1) * x(2) ... x(n)
(e.g. x^3 = 1 * x * x * x)

It follows that

x^1 = 1 * x and x^0 = 1

The latter case is called the empty product because every multiplication has an implied 1. Since we removed all the terms (as implied by the zeroth power), we're just left with that 1.

I am not sure I like this statement. As an educator, if I were going to teach this as a rule, I would certainly want to explain why every multiplication has an "implied 1". I don't have a good answer for that, do you?

And I am curious... under this ideology, does every addition have an implied zero?

"If you want to pass the test, remember that anything raised to the power of zero equals 'one'."

I saw straight lines in the sky just the other day. Sunbeams coming through the clouds.

Well, why every multiplication has an implied 1 is rather simple to conceptualize. Multiplication is basically "quantity x of quantity y." I'm pretty sure we can all agree that if you have 1 of a quantity, you have that quantity. If I, say, had 1 group of 8 apples, well, then I have those 8 apples. If I had 1 group of 3.5 bananas, I'd have 3.5 bananas.

Now, this extends to general multiplications. Say I have 1 set of 5 sets of 9 bananas each. Well, in the 5 sets of 9 bananas each, I'd have a total of 45 bananas, right? Now, if I had just 1 of this superset, then I'd still have a set of 45 bananas. This all is the basis for the multiplicative identity.

As for your question regarding every addition having an implied zero ... well, it's a similar (but not equivalent) principle: when you add nothing to something, you end up with what you had to start with. Add 0 to 2 and you end up with 2. Add 0 to pi and you end up with pi. If I had 8 oranges and someone gave me none, I still have 8 oranges. This is otherwise known as the additive identity.

Hope this helps.

EDIT: By this same reasoning, we say that all numbers are complex numbers, just that the real numbers are complex numbers of the form a + 0i, with a null imaginary part. But I'm getting ahead of myself, here.

Huh? This isn't simple to conceptualize at all. I mean I understand why multiplying by one gives me the same number-- it is easy to understand why one group of 8 apples equals 8 apples.

It is also easy to understand why 4 groups of 8 apples equals 32 apples.

In the first case, I understand why the number 1 is involved. In the second example the numbers invlued are 4 and 8.... you haven't at all explained why the number 1 has anything to do with it.

Of course 4 apples is really 2 groups of 2 apples. Does this mean there is an implicit 2 in the multiplication of 4 * 8?

If you have a total of 32 apples, you have a set of 32 apples, do you not? That's where the implied 1 comes in. 1 * 32 = 32.

4 * 8 does not have an implicit 2; it is very explicit. It can be factored as 2^2 * 2^3 = 2^5. But there's the multiplicative identity, which makes that actually 1 * 2^5 = 1 * 2 * 2 * 2 * 2 * 2.

When you have a quantity, you always have a quantity ... 1 of that quantity. That's where the implied 1 comes from.

I multiply 25 toothpicks by 4 to get 100, I end up with a set of 100 toothpicks, 1 * 100 = 100. The 1 is always there.

Except when you're asked to factor 32, you don't say "2^5*1^∞"; just as you don't say 5 is 2+3+0*∞.

No. Tokyo... you are not making things simpler. You are making them more difficult. And you still haven't proven (or even explained) your arbitrary "implicit one" thing.

First; noticed how you have changed . First you said there is an "implicit one" in every multiplication (as if this were some mysitical truth about the act of multiplying). Now you are saying ther is an "implicit one" in every "quantity".

This is an interesting difference. With an "implicit one" in multiplication you would change 4 * 8 = 32 into 1 * 4 * 8 = 32.

With your latest argument this would really turn into 1 * 4 * 1 * 8 = 32 (or would that be 1 * 4 * 1 * 8 = 1 * 32).

Of course this leads to the funny fact that each of these implicit ones are also quantities leading to --- 1 * 1 * 4 * 1 * 1 * 8 = 1 * 1 * 32 (and of course this leads to a never ending process of mindless expansion). Of course this is all arithmetically correct, but I fail to see how this has any value in teaching kids about the underlying mathematics.



This is also completely true, but educationally meaningless (other than the property of multiplicative identity).

I could just as easily point out that when you have a quantity, you always have twice the value of half of that quantity---

... this is more proof that every multiplication has an implicit 2.

It doesn't matter. It's a simple truth that the 1 is always there. XD

No, it's there only when you decide to put it there. The identity property equates two things, but being equated does not make them the same thing.

Cos(0)=Sin(90)=Tan(45)=1=x^0, but these are all different things.

I do not see where more explanation is needed. The multiplicative identity should be intuitive enough. It's how my math teacher explained it to me (more or less), and I have no problems understanding it.

Your question about the "implied 2" ultimately falls back to that implied 2 being nestled in a value of 1, or 2/2. Of course, you could make an argument for any (nonzero) number being implied this way. Unfortunately, it is an equivocation of my original "implied 1" statement, as it introduces a step (the division) that makes the two statements incomparable.

...

Okay, if you insist.

If you just swallow the idea of "implicit 1" because your teacher told you this as a rule-- and never questioned it or thought about what it means, than your teacher has done you a great disservice.

This is the real issue here. By giving these simplistic "rules of thumb", you cover over the real concepts involved. At the very least, you should appreciate that now you are having to explain this rule that you seem to have accepted as a given.



Funny... you haven't given an argument for this mystical "implied 1" that I can't turn into an argument for an "implied 2".

It is unquestionable a rule as the rule that you can't divide by zero. It is an axiom; there is no escaping it with critical thinking, because it is one of the basic rules we use to think about everything else. And you continually end up falling back on the 2 being hidden in a value of 1 with blatantly fallacious reasoning.

It is completely ridiculous to say that anything is unquestionable in mathematics. Questioning is the fundamental activity of mathematics... and if someone asks you to accept something without question, they aren't doing mathematics.

If you ask why you can't divide by zero, we will have a very interesting conversation (depending on your level of mathematics). Of course, there are plenty of reasons that, after asking the questions, we will reach the conclusion that you can't divide by zero.

If you haven't questioned the rule about dividing by zero--- then your mathematics education is clearly lacking. Mathematics is not a set of rules to learn and accept. It is a language for investigation and understanding. If you can't explain why a concept is valid-- then you don't really understand the concept. Reaching this understanding is a process of questioning.

You keep saying-- "there is an implied one because there is an implied one"-- not much of a proof, and not a very mathematical argument. I am simply asking "why"?